Finite Group Databasebeta

← #10828640

Subgroups — #10828640

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85746C211yesmaximal, characteristic
2C3×#2285C211yesmaximal, characteristic, complemented
3#82216C212yesmaximal, complemented
4#71589C211yesmaximal, characteristic, complemented
5#71532C212yesmaximal, complemented
6#69593C211yesmaximal, characteristic, complemented
7#69592C212yesmaximal, complemented
8#69573C211yesmaximal, characteristic, complemented
9#69563C211yesmaximal, characteristic, complemented
10#69562C212yesmaximal, complemented
11#69543C211yesmaximal, characteristic, complemented
12#24078—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16#5791C2212yes—
17#5533—22no—
18#5532—22no—
19#5530—22no—
20#5529—22no—
21#5510C2211yescharacteristic
22#5504C2212yes—
23#5500C2211yescharacteristic
24#5494—22no—
25#5479—22no—
26C3×#384C2211yescharacteristic
27C3×#384C2211yescharacteristic
28C3×#382C2211yescharacteristic
29C3×#382C2211yescharacteristic
30C3×#380—22no—
31C3×C4×C4⋊C4—22no—
32#5207C2212yes—
33#5003—22no—
34#5003—22no—
35#4996—22no—
36#4996—22no—
37#4966C2212yes—
38#4966C2212yes—
39#4964C2212yes—
40#4964C2212yes—
41C2×#805C2212yes—
42C2×#805C2212yes—
43#4530C2211yescharacteristic
44#4530C2212yes—
45#4530C2211yescharacteristic
46#4523C2212yes—
47#4523C2212yes—
48#3493—31no—
49#3093—32no—
50#2285D311yescharacteristic, complemented
51#2061—31no—
52#2060—31no—
53#2040—31no—
54#2039—31no—
55#1896—32no—
56#1860—32no—
57#1859—32no—
58#1822—31no—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C2×#235C2312yes—
62C3×#117—22no—
63C3×#117—22no—
64C3×#117—22no—
65C3×#117—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4D412yes—
74C2×C3×C42—21no—
75C2×C3×C42—22no—
76C2×C3×C42—22no—
77C2×C3×C42—21no—
78C2×C3×C42D412yes—
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81#953—22no—
82#930—22no—
83#930D412yes—
84#930D412yes—
85#929—22no—
86C4×#67—24no—
87C4×#67—22no—
88C4×#67—22no—
89#926C2312yes—
90#924—22no—
91#924D412yes—
92#924D412yes—
93C2×#208C2312yes—
94C2×#208C2312yes—
95#897—22no—
96#895—22no—
97#893—22no—
98#892—22no—
99D3×C4⋊C4—22no—
100#890—22no—
101#888—22no—
102#887—22no—
103#885—24no—
104#884—24no—
105#883—24no—
106#879—24no—
107#876—24no—
108#875—22no—
109C4×D12—22no—
110#872—24no—
111#870—22no—
112C4×Dic6—22no—
113C3×#95—22no—
114C3×#95—22no—
115#830—22no—
116#830—22no—
117#810—22no—
118#810—22no—
119#809—22no—
120#809—22no—
121#808—22no—
122#808—22no—
123#807—22no—
124#807—22no—
125#805—24no—
126#805—24no—
127#556—62no—
128#555—62no—
129#553—62no—
130#552—62no—
131#544—31no—
132#538—62no—
133#537—31no—
134#529—62no—
135#525—32no—
136#520—32no—
137C2×#117C2×D311yescharacteristic
138C2×#117C2×D311yescharacteristic
139C22×C42C2×D311yescharacteristic
140#485—62no—
141#485—62no—
142#482—62no—
143#482—62no—
144#428—32no—
145#428—32no—
146C2×#104—32no—
147C2×#104—32no—
148#417—32no—
149#417—32no—
150#407—32no—
151#384C2×D311yescharacteristic
152#384C2×D311yescharacteristic
153#382C2×D311yescharacteristic
154#382C2×D311yescharacteristic
155#380—22no—
156C4×C4⋊C4—22no—
157#343—31no—
158#343—31no—
159#343—32no—
160#339—32no—
161#339—32no—
162C24×C3C2×D411yescharacteristic
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—42no—
172C22×C3×C4—42no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2411yescharacteristic
179C2×#67—22no—
180C2×#67—22no—
181C2×#67—22no—
182#235—22no—
183#235—22no—
184#235—22no—
185#235—22no—
186C2×D12—22no—
187C2×D12C2×D412yes—
188C2×C4×D3—22no—
189C2×C4×D3—24no—
190C2×Dic6—22no—
191C2×Dic6C2×D412yes—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—44no—
197C3×C4⋊C4—44no—
198C3×C22⋊C4—24no—
199C3×C22⋊C4—42no—
200C3×C42—24no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—21no—
204C3×C42—21no—
205C3×C42—41no—
206C3×C42—41no—
207C3×C42#4112yes—
208C3×C42#4112yes—
209#217—22no—
210#217—22no—
211#217—22no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—22no—
215(C2×D3)⋊C4—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3C2×D412yes—
219C4⋊Dic3C2×D412yes—
220Dic3⋊C4—24no—
221Dic3⋊C4—24no—
222Dic3⋊C4—22no—
223Dic3⋊C4—22no—
224C4×Dic3—24no—
225C4×Dic3—22no—
226#208—22no—
227#208—22no—
228#208—22no—
229#208—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×#41—32no—
233C23×C4—21no—
234C23×C4C22×D311yescharacteristic
235C2×M16—32no—
236C2×M16—32no—
237C4⋊Q8—62no—
238#126—64no—
239#126—64no—
240#125—62no—
241#125—62no—
242#124—62no—
243#124—64no—
244#123—64no—
245#123—62no—
246C22⋊Q8—32no—
247C22⋊Q8—32no—
248C22⋊Q8—62no—
249C22⋊Q8—62no—
250#121—32no—
251#121—32no—
252#121—64no—
253#121—62no—
254#121—62no—
255#120—62no—
256C4×Q8—62no—
257C4×Q8—62no—
258C4×D4—62no—
259C4×D4—62no—
260C4×D4—62no—
261C4×D4—62no—
262C4×D4—64no—
263#117—22no—
264#117—22no—
265#117—22no—
266#117—22no—
267#117—32no—
268#117—64no—
269#117—62no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4#6712yes—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42#6712yes—
284C2×C42C22×D311yescharacteristic
285C2×C42C22×D311yescharacteristic
286#107—62no—
287#107—62no—
288#106—62no—
289#106—62no—
290#104—64no—
291#104—64no—
292#103—62no—
293#103—62no—
294#102—62no—
295#102—62no—
296C22⋊C8—62no—
297C22⋊C8—62no—
298#95—22no—
299#95—22no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3#12012yes—
305C23×C3C22×D411yescharacteristic
306C22×D3—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—21no—
310C2×C3×C4—22no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—48no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4#12011yescharacteristic
329C2×C3×C4#12011yescharacteristic
330C2×C3×C4C22×D412yes—
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C42−1+412yes—
334#67—42no—
335#67—42no—
336#67—82no—
337#67—82no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341D12—22no—
342D12—22no—
343D12—82no—
344C4×D3—42no—
345C4×D3—42no—
346C4×D3—84no—
347Dic6—22no—
348Dic6—22no—
349Dic6—82no—
350C3⋊C8—42no—
351C3⋊C8—42no—
352C24C2×#6711yescharacteristic
353#41—62no—
354#41—62no—
355#41—62no—
356#41—62no—
357C2×Q8—32no—
358C2×Q8—62no—
359C2×D4—32no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—62no—
363C2×D4—62no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—64no—
375C22×C4—62no—
376C22×C4D3×D412yes—
377C22×C4D3×D412yes—
378C22×C4C2×#6711yescharacteristic
379C22×C4C2×#6712yes—
380C22×C4C2×#6712yes—
381C22×C4C23×D311yescharacteristic
382M16—62no—
383M16—62no—
384M16—62no—
385M16—62no—
386C2×C8—62no—
387C2×C8—62no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—32no—
393C4⋊C4—32no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—64no—
397C4⋊C4—64no—
398C4⋊C4—62no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C22⋊C4—24no—
403C22⋊C4—42no—
404C22⋊C4—64no—
405C22⋊C4—64no—
406C22⋊C4—62no—
407C22⋊C4—62no—
408C22⋊C4—62no—
409C22⋊C4—62no—
410C22⋊C4—62no—
411C42—24no—
412C42—22no—
413C42—22no—
414C42—21no—
415C42—21no—
416C42—41no—
417C42—41no—
418C42—64no—
419C42—62no—
420C42#23512yes—
421C42#23512yes—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3C2×#12011yescharacteristic
430C22×C3#52511yescharacteristic
431C22×C3#53611yescharacteristic
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—42no—
435C3×C4—24no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—42no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—81no—
442C3×C4—81no—
443C3×C4#54712yes—
444C3×C4#54712yes—
445Dic3—42no—
446Dic3—42no—
447Dic3—42no—
448C23—21no—
449C23—21no—
450C23—41no—
451C23—41no—
452C23—62no—
453C23#93712yes—
454C23C22×#6711yescharacteristic
455Q8—62no—
456Q8—62no—
457Q8—242no—
458D4—62no—
459D4—62no—
460D4—122no—
461D4—122no—
462D4—242no—
463D4—242no—
464D4—242no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—48no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—244no—
492C2×C4#93711yescharacteristic
493C2×C4#93711yescharacteristic
494C2×C4C2×#23511yescharacteristic
495C2×C4C2×#23511yescharacteristic
496C2×C4C2×D3×D412yes—
497C2×C4#100712yes—
498C8—122no—
499C8—122no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—42no—
503C2×C3#240611yescharacteristic
504C2×C3#301811yescharacteristic
505C2×C3#301811yescharacteristic
506D3—82no—
507C22—21no—
508C22—21no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—42no—
513C22—42no—
514C22—122no—
515C22—122no—
516C22—122no—
517C22#579111yescharacteristic
518C22C2×#93711yescharacteristic
519C22#581611yescharacteristic
520C4—24no—
521C4—21no—
522C4—21no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—81no—
527C4—81no—
528C4—122no—
529C4—122no—
530C4—122no—
531C4#550712yes—
532C4#550712yes—
533C3#2407811yescharacteristic, complemented, Sylow
534C2—21no—
535C2—21no—
536C2—42no—
537C2—242no—
538C2#8219211yescharacteristic
539C2#8332211yescharacteristic
540C2#8332211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2285C2
socleC22×C3C2×#120
prime radical#85746C2
hypercenter#2285D3
hypocenterC3#24078
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2285

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#2285C2semidirect
3. #82216C2semidirect
4. #71589C2semidirect
5. #71532C2semidirect
6. #69593C2semidirect
7. #69592C2semidirect
8. #69573C2semidirect
9. #69563C2semidirect
10. #69562C2semidirect
11. #69543C2semidirect
50. #2285D3semidirect
533. C3#24078semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#857462
C3×#22852
#822162
#715892
#715322
#695932
#695922
#695732
#695632
#695622
#695432
#240783